Small Noise Asymptotics of the Bayesian Estimator in Nonidentifiable Models

We study the asymptotic behavior of the Bayesian estimator for a deterministic signal in additive Gaussian white noise, in the case where the set of minima of the Kullback–Leibler information is a submanifold of the parameter space. This problem includes as a special case the study of the asymptotic behavior of the nonlinear filter, when the state equation is noise-free, and when the limiting deterministic system is nonobservable. As the noise intensity goes to zero, the posterior probability distribution of the parameter asymptotically concentrates on the submanifold of minima of the Kullback–Leibler information. We give an explicit expression of the limit, and we study the rate of convergence. We apply these results to a practical example where nonidentifiability occurs.